{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Monty Hall Problem"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem Description:\n",
    "The Monty Hall Problem is a very famous problem in Probability Theory. The question goes like:\n",
    "\n",
    "\n",
    "Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, \"Do you want to pick door No. 2?\" Is it to your advantage to switch your choice?\n",
    "\n",
    "By intution it seems that there shouldn't be any benefit of switching the door. But using Bayes' Theorem we can show that by switching the door the contestant has more chances of winning.\n",
    "\n",
    "You can also checkout the wikipedia page: https://en.wikipedia.org/wiki/Monty_Hall_problem"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Probabilistic Interpretetion:\n",
    "So have 3 random variables Contestant $C \\in \\{1, 2, 3\\}$, Host $H \\in \\{1, 2, 3\\}$ and prize $P \\in \\{1, 2, 3 \\}$. The prize has been put randomly behind the doors therefore: $P(P=1) = P(P=2) = P(P=3) = \\frac{1}{3}$. Also, the contestant is going to choose the door randomly, therefore: $P(C=1) = P(C=2) = P(C=3) = \\frac{1}{3}$. For this problem we can build a Bayesian Network structure like:\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import Image\n",
    "\n",
    "Image(\"images/monty.png\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "with the following CPDs:\n",
    "\n",
    "<pre>\n",
    "\n",
    "P(C):\n",
    "+----------+----------+-----------+-----------+\n",
    "|    C     |     0    |     1     |      2    |\n",
    "+----------+----------+-----------+-----------+\n",
    "|          |    0.33  |    0.33   |    0.33   |\n",
    "+----------+----------+-----------+-----------+\n",
    "\n",
    "P(P):\n",
    "+----------+----------+-----------+-----------+\n",
    "|    P     |     0    |     1     |      2    |\n",
    "+----------+----------+-----------+-----------+\n",
    "|          |    0.33  |    0.33   |    0.33   |\n",
    "+----------+----------+-----------+-----------+\n",
    "\n",
    "P(H | P, C):\n",
    "+------+------+------+------+------+------+------+------+------+------+\n",
    "|   C  |          0         |          1         |          2         |\n",
    "+------+------+------+------+------+------+------+------+------+------+\n",
    "|   P  |   0  |   1  |   2  |   0  |   1  |   2  |   0  |   1  |   2  |\n",
    "+------+------+------+------+------+------+------+------+------+------+\n",
    "|  H=0 |   0  |   0  |   0  |   0  |  0.5 |   1  |   0  |   1  |  0.5 | \n",
    "+------+------+------+------+------+------+------+------+------+------+\n",
    "|  H=1 |  0.5 |   0  |   1  |   0  |   0  |   0  |   1  |   0  |  0.5 |\n",
    "+------+------+------+------+------+------+------+------+------+------+\n",
    "|  H=2 |  0.5 |   1  |   0  |   1  |  0.5 |   0  |   0  |   0  |   0  |\n",
    "+------+------+------+------+------+------+------+------+------+------+\n",
    "</pre>\n",
    "\n",
    "Let's say that the contestant selected door 0 and the host opened door 2, we need to find the probability of the prize i.e. $P(P|H=2, C=0)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Finding Elimination Order: : : 0it [00:19, ?it/s]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<TabularCPD representing P(C:3) at 0x7f580a175310>,\n",
       " <TabularCPD representing P(P:3) at 0x7f58128ad520>,\n",
       " <TabularCPD representing P(H:3 | C:3, P:3) at 0x7f580a175340>]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from pgmpy.models import BayesianNetwork\n",
    "from pgmpy.factors.discrete import TabularCPD\n",
    "\n",
    "# Defining the network structure\n",
    "model = BayesianNetwork([(\"C\", \"H\"), (\"P\", \"H\")])\n",
    "\n",
    "# Defining the CPDs:\n",
    "cpd_c = TabularCPD(\"C\", 3, [[0.33], [0.33], [0.33]])\n",
    "cpd_p = TabularCPD(\"P\", 3, [[0.33], [0.33], [0.33]])\n",
    "cpd_h = TabularCPD(\n",
    "    \"H\",\n",
    "    3,\n",
    "    [\n",
    "        [0, 0, 0, 0, 0.5, 1, 0, 1, 0.5],\n",
    "        [0.5, 0, 1, 0, 0, 0, 1, 0, 0.5],\n",
    "        [0.5, 1, 0, 1, 0.5, 0, 0, 0, 0],\n",
    "    ],\n",
    "    evidence=[\"C\", \"P\"],\n",
    "    evidence_card=[3, 3],\n",
    ")\n",
    "\n",
    "# Associating the CPDs with the network structure.\n",
    "model.add_cpds(cpd_c, cpd_p, cpd_h)\n",
    "\n",
    "# Some other methods\n",
    "model.get_cpds()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# check_model check for the model structure and the associated CPD and returns True if everything is correct otherwise throws an exception\n",
    "model.check_model()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Finding Elimination Order: : : 0it [00:00, ?it/s]\n",
      "0it [00:00, ?it/s]\u001b[A"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "+------+----------+\n",
      "| P    |   phi(P) |\n",
      "+======+==========+\n",
      "| P(0) |   0.3333 |\n",
      "+------+----------+\n",
      "| P(1) |   0.6667 |\n",
      "+------+----------+\n",
      "| P(2) |   0.0000 |\n",
      "+------+----------+\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\n"
     ]
    }
   ],
   "source": [
    "# Infering the posterior probability\n",
    "from pgmpy.inference import VariableElimination\n",
    "\n",
    "infer = VariableElimination(model)\n",
    "posterior_p = infer.query([\"P\"], evidence={\"C\": 0, \"H\": 2})\n",
    "print(posterior_p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see that the posterior probability of having the prize behind door 1 is more that door 0. Therefore the contestant should switch the door."
   ]
  }
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